Discussion on boundary controllability of nonlocal fractional neutral integrodifferential evolution systems

1.   Introduction

Most of the dynamical systems in science and engineering have the intrinsic nature of time delay to some degree. Time delays occur in many dynamical systems such as chemical or process control systems, biological systems, network control systems, etc. As per the practical requirement, the delays may be time-varying or constant. Many systems, for example, sample data control, aircraft control, etc., are modelled as time-varying delays, which are the principal cause of instability and weak control performance. Neutral systems with time delay are the systems in which uncertainty is present in the state and derivatives. These systems arise in many areas like population ecology, distributed networks, etc. Due to various applications in engineering and science, many researchers worked on the delay differential equation. Besides the above, in control theory, the problem of controllability is of specific importance. Controllability of the system can handle various problems like stabilization, pole assignment, and optimal control.

Fattorini ^[1] introduced the controllability condition on the boundary control system of first and second-order replacing boundary controls by distributed controls. Park et al. ^[2,4] gave the results for the boundary controllability of semilinear nonlocal control systems with the help of the Banach fixed point theorem. Ahmed et al. ^[3] derived results for approximate boundary controllability of stochastic control system of fractional order having Poisson jump and fractional Brownian motion. Balachandran et al. ^[5] investigated sufficient conditions for controllability of abstract neutral integrodifferential control systems having infinite delay using Nussbaum fixed point theorem and analytic semigroup theory. Radhakrishnan et al. ^[7] and Kumar et al. ^[10] studied the controllability for the integrodifferential control system in Banach space with the help of the Schaefer fixed point theorem. Cheng et al. ^[11] presented the exact controllability of fractional control system having time-varying delay using the theory of propagation family and Leray-Schauder theorem. Applications of fractional calculus in neural networks can be found in ^{[45,46,47,48,49,50,51,52,53,54]}.

In ^{[12,13,14,15,16,23]} researchers investigated the existence and uniqueness of mild solution for certain classes of a fractional evolution equation with the help of fixed point theorem, fractional power operators, semigroup theory, the measure of non-compactness, etc. Guo et al. ^[21] presented null boundary controllability and proved nonlinear Cauchy-Kowalevski theorem for 1-D semilinear heat equations. In ^{[8,9,17,18,19,20,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,44]}, researchers studied the approximate controllability for the semilinear control system of fractional order $\alpha \in (1, 2]$, first and second-order using fixed-point theorem, cosine and sine theory of operators, fractional calculus, Lipschitz continuity and sequential approach.

Under Lion boundary conditions, the 3D Navier-Stokes system is approximately controllable, as discussed by Phan et al. ^[40]. Meraj et al. ^[41] derived the results for approximate controllability for a non-autonomous control system having nonlocal initial conditions using Krasnoselski theorem and evolution system. In modern era, fixed point theory is employed in many areas of mathematics and engineering. Keeping these utilizations of fixed point theory in mind, some fixed point results are proved in the structure of partial $b$-metric spaces and allied abstract spaces which emphasize primarily the applications for existence of the solution of various functional equations occurring in dynamic programming, integral equations, boundary value problems, equations representing LCR circuits and simple harmonic motion. Fixed point theory is a powerful tool to determine uniqueness of solutions to dynamical systems and is widely used in theoretical and applied analysis.

The main contributions of our paper are:

● We have derived the results for the boundary controllability of integrodifferential neutral control system having time-varying delays of fractional order.

● The primary outcomes for the systems (2.1)–(2.3) are derived by employing Banach fixed point theorem, semigroup theory and fractional calculus.

● By considering a set of assumptions $(A1)$–$(A10)$ results can be obtained.

● The research focused on the boundary controllability of proposed systems (2.1)–(2.3) under consideration are not addressed in the literature to our knowledge, and it supports the current findings.

This article's structure will now be presented as follows:

(1) Section 2 discusses some definitions and assumptions.

(2) Results for boundary controllability are presented in section 3.

(3) An example is demonstrated to validate the theoretical result.

(4) Finally we have given a conclusion section.

2.   Preliminaries

Let us assume that $V$ and $U$ are two real Banach spaces having norms $\|\cdot\|$ and $|\cdot|$ respectively. Consider that $\rho$ be a linear, closed and densely defined operator with $D(\rho) \subseteq V$. Consider $\Theta \subseteq X$ be linear operator with $D(\rho)$ and $R(\Theta) \subseteq X$, a Banach space. Now define the nonlocal fractional integrodifferential boundary control system having time varying delay as:

where $\varsigma_1 > 0$, $\varsigma_p \leq b$ and $\varsigma_i < \varsigma_{i+1}\; \; i = 1, 2, .....p-1$ and $\alpha \in (0, 1]$. The continuous linear operator $E_1$ is defined from $U$ to $X$. The control function $u\in L^2(J, U)$, a Banach space of admissible control functions with $U$. The delays functions $\gamma_1, \gamma_2$ and $\gamma_3$ are continuous. The state $s(.) \in V$, the functions $g, \phi, \Psi$ and $\eta$ are defined as $g:J\times V\times V \rightarrow V$, $\phi:J^p\times V\rightarrow V$, $\Psi:\Delta \times V\rightarrow V$ and $\eta:J\times V \times V \rightarrow V,$ and $\Delta = \{(\varsigma, \varrho), 0 \le \varrho \le \varsigma \le b\}$.

Suppose that the linear operator $A:D(A) \rightarrow V$ where $D(A) \subseteq V$ be defined as: $D(A) = \{s\in D(\rho) : \Theta s = 0\}, As = \rho s$, for $s \in D(A)$. Consider

We consider the following assumptions:

$(A_1)\; D(\rho)\subset D(\Theta)$ where $\rho$ is closed linear operator having domain $D(\rho)$ and $\Theta$ is a boundary operator having domain $D(\Theta)$. Also the restriction of $\Theta$ to $D(\rho)$is continuous (w.r.t. graph norm).

$(A_2)\; T(\varsigma)$ is a compact semigroup having infinitesimal generator $A$. Also there exist $L \in (0, \infty)$ satisfying

$(A_3)$ There exists an operator $E$ defined from $U$ to $V$ which is continuous and linear, with $\rho E\in L(U, V)$, $\Theta(EU) = E_1U$, $\forall u\in U$. In addition, there exist a constant $C > 0$ such that $\|\ Eu \|\leq C\|\ E_1u \|$ for every $u\in U$ and continuous differentiability is satisfied by $Eu(\varsigma)$.

$(A_4)$ For $u \in U$ and $0 < \varsigma \leq b$, $T(\varsigma)Eu \in D(A)$. Also, there is a constant $M \in (0, \infty)$ satisfying $\|\ AT(\varsigma) \| \le M$.

$(A_5)$ The operator $W:L^2(J, U)\rightarrow V$ is linear and explained in the following way

where $\xi_{\alpha} (\varpi)$ is called p.d.f, $0 < \varpi < \infty$, see ^[13,14]. The operator $\tilde W^{-1}$ induced by $\xi_{\alpha} (\varpi)$, is invertible and is defined on $L^2(J, U)/ ker W$. There exist $L_1, L_2 \in (0, \infty)$ satisfying

and

$(A_6)$ The function $\eta$, $g$ is continuous in $\varsigma$. Also, for every $s_i, y_i, u_i, v_i \in E_r, \; \; i = 1, 2$ and $\varsigma \in J$, there exist $C_1, M_1 \in (0, \infty)$

$(A_7)$ The function $\Psi$ is continuous in $\varsigma$ and $\varrho$, there exist $N_1 \in (0, \infty)$ such that for every $u, v \in E_r$ and $\varsigma, \varrho \in \Delta$,

$(A_8)$ There exists a constant $H\in (0, \infty)$ such that for $z_1, z_2 \in C(J, E_r)$, we have

$(A_{9})\; \forall s_1, s_2 \in V$ there is a constant $p > 0$ such that

$(A_{10})$ There exists a constant $r > 0$ such that

where the constants $M_2, C_2, C_3, C_4$ and $N_2$ will be defined later.

Let us assume the solutions of (2.1)–(2.3) as $s(\varsigma)$. Now, describe $z(\varsigma) = -Eu(\varsigma)+s(\varsigma)$ and by above assumption, $z(\varsigma) \in D(A)$.

We can express (2.1)–(2.3) as:

If continuous differentiability on $[0, b]$ is satisfied by $u$, then we can express $z$ as a mild solution of given below problem:

The solutions of (2.1)–(2.3) is stated as:

Now, integrating (2.7), we get

Consequently, (2.8) is clearly defined and it is called a mild solution of the systems (2.1)–(2.3).

Definition 2.1. The systems (2.1)–(2.3) is called controllable on the interval $J$ if for all $s_0, a \in V$, there exists a control $u \in L^2(J, U)$ such that the solutions $s(\cdot)$ of (2.1)–(2.3) meets $s(b) = a$ where $b > 0$ is a final point of $J$.

The main objective is to transfer the systems (2.4)–(2.6) from $s(0) = s_0 -\phi(\varsigma_1, \varsigma_2, ..., \varsigma_p, s(\cdot))$ to $s(b) = a$, see ^[4].

3.   Main results

Theorem 3.1. If the assumptions $(A_1)$–$(A_{10})$ are fulfilled, then the boundary control of the evolution systems $(2.1)$–$(2.3)$ is controllable on $J$ provided

Proof. By applying the invertible operator $\tilde W^{-1}$, for arbitrary function $s(\cdot)$, we present

Now, we shall observe that, when applying this control $u(\varsigma)$, the operator

given as

has a fixed point. This fixed point is then the solutions of the (2.1)–(2.3). $Fx(b) = a$, this means that the control $u$ steers the nonlocal fractional neutral integrodifferential evolution systems with time varying delays from the initial state $s_0$ to $a$ in time $b$ provided we can achieve a fixed point of the operator $F$.

Initially, we demonstrate that $F$ maps $C(J; E_r)$ into itself. We can select $M_2, C_2, C_3,$$C_4, N_2 > 0$ such that

Moreover, in view of the fact $s(\cdot)$ in $\phi$ is continuous on $J$, we consider

From Eq (3.1), we have

Hence, $F$ maps $C(J; E_r)$ into itself.

Next, we demonstrate that $F$ is a contraction on $C(J; E_r)$. Indeed,

where

Since $0\le \Pi < 1$, $F$ is a contraction on $C([0, b]; E_r)$. Implementing the Banach fixed point theorem we obtain a unique fixed point for $F$ in $C(J; E_r)$ and which is the mild solutions of (2.1)–(2.3) that fulfills $s(b) = a$. Thus, the systems (2.1)–(2.3) is controllable on $J$.

4.   Application

Let $\Omega$ be a bounded subset of $\mathbb{R}^{n}$ and let boundary of $\Omega$ be $\Gamma$, which is smooth. Consider the boundary control nonlocal neutral fractional differential system with time varying delay

in which $u\in L^2(\Sigma), z_0 \in L^2(\Omega), g, \Psi\in L^2(Q)$.

Moreover, consider that the functions $\eta, g, \Psi$ and $\phi$ are fulfilled the conditions given as:

and

where $a_1, a_2, a_3, a_4$ are positive constants, $s, y, z, u, v \in E_r$ and $s_1, s_2 \in C(J; E_r)$. Now, we can take operators $\rho$, $E_1$ and $\Theta$, the space $U$, $X$, $V$ as below, so we can formulate (4.1)–(4.3) as a boundary control problems of (2.1)–(2.3).

$\rho = \Delta,$ $E_1$ is equal to $I$, where $I$ denotes identity operator. $\Theta$ is called trace operator and is defined as $\Theta z = z|_{\Gamma}$. Clearly $\Theta$ is well defined and $\Theta \in H^{-\frac{1}{2}}(\Gamma)$.

Also, $L^2(\Gamma) = U$, $X = H^{-\frac{1}{2}}(\Gamma)$ and $V = L^2(\Omega)$. $A$ is defined as $A = \Delta, D(A) = H^1_0(\Omega)U H^2(\Omega)$, where $H^k(\Omega), H^{\chi}(\Gamma)$ and $H^1_0(\Omega)$ are usual Sobolev space on $\Omega, \Gamma$.

Then $A$ can be represented as:

where $z_n(y) = \sqrt{2} \sin ny, \; n = 1, 2, ....$ is the orthogonal set of eigenvectors of $A$.

Again, for $z\in U$,

It is easy to see that $A$ generates a strongly continuous semigroup $\{T(\varsigma):\varsigma \geq 0\}$ in $U$. Hence, the hypothesis $(A1), (A2)$ are satisfied.

We express the operator $E$ defined from $L^2(\Gamma)$ to $L^2(\Omega)$ as $Eu = w_u$, $w_u$ is the unique solution to the subsequent system,

It is proved in ^[22] that for every $u \in H^{-\frac{1}{2}}(\Gamma)$, the above system has a unique solution $w_u \in L^2(\Omega)$ satisfying $||Eu||_{L^2(\Omega)} = ||wu||_{L^2(\Omega)} = c_1 ||u||_{H^{-\frac{1}{2}}(\Gamma)}$. This shows that (A3) is satisfied. From the above estimates, it follows by interpolation argument ^[6] that

with $v(\varsigma) = C' \varsigma^{\frac{-3}{4}},$ where $c_1, C'$ are positive constant independent of $u$. Therefore, hypothesis (A4) is satisfied. The boundary controllability of the systems (4.1)–(4.3) is discussed in detailed in ^[1,5]. For condition (A5), the detailed discussion is presented in ^[42,43]. Clearly, the functions $\eta, g, \Psi, \phi$ satisfies the assumptions (A6)–(A9).

Select $b$ and the remaining constants such that $(A_1)$–$(A_{10})$ are fulfilled, see ^[22]. Thus, Theorem 3.1 can be applied for (4.1)–(4.3) and so the systems (4.1)–(4.3) is controllable on $[0, b]$.

5.   Conclusions

To sum up it, we explored the sufficient conditions for boundary controllability of neutral integrodifferential evolution system of fractional order with time varying delay and nonlocal condition in Banach spaces. To establish the result, we apply the Banach fixed point theorem. In the end, we stated an application to validate the abstract result. The above researches of (2.1)–(2.3) provides only an analytic result. This result which is given in (2.1)–(2.3), may be applied to get the numerical solution of such kind of equation. In future we will try to obtain some results on stability, asymptotic behavior and some numerical method like HAM. Also researchers can extend boundary controllability to boundary communication.

Acknowledgments

This work was supported by the research grant of Kongju National University in 2021 and this research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (No. 2021R1F1A1048937) to the first author.

Conflict of interest

This work does not have any conflict of interest.